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' ~MzIID>0 d2BJBXz mI E ,$D  0' F B>?"6@ NNN?Nme oMzI6>0 d2BJ5 G Bc>?"6@ NNN?N+hI }SzID>0 d2BJBB H BD>?"0@NNN?N>EB I HsD>?"0@NNN?N~lsz p J 66,$D 02 K H jJ?"6@ NNN?NC2 LB H jJ?"6@ NNN?NpJ d+ ^  M# |YG / ,$D! 02 N N! jJ?"6@ NNN?N]+ ^ 2 O T jJ?"6@ NNN?Nd e  d+ ^  P# o d,$D# 02 Q H! jJ?"6@ NNN?N]+ ^ 2 R T jJ?"6@ NNN?Nd e z p S /,$D 02 T H jJ?"6@ NNN?NC2 UB H jJ?"6@ NNN?NpJ V Tl_ж_ж?"`  EInternal forcesF W Bp Ԕ?"6@ NNN?N  ,$D' 0 {wII} a" {ZI} >0 d2A@HAaA@HA  X Bx Ԕ?"6@ NNN?N . ,$D& 0 {wI} a" {ZII} >0 d2A@HAaA@HA a  H  0޽h ?KLNOQRTU 33XW___PPT10W.֝``+zD&V' 6= @B DU' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(DF' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*=%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*>%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*?%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D' =4@BB#BB%()?)?D$' =.7 BBBBB)M 0 0 L -0.25 0 E*3>*B ppt_xB ppt_y=0BB<*D' =4@BB#BB%()?)?D$' =.7 BBBBB)M 0 0 L -0.25 0 E*3>*B ppt_xB ppt_y=0BB<*$D' =4@BB#BB%()?)?D$' =.7 BBBBB)M 0 0 L -0.25 0 E*3>*B ppt_xB ppt_y=0BB<*D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =4@BB?BB%()?)?D"' =.7 BBBBB'M 0 0 L 0.25 0 E*3>*B ppt_xB ppt_y=0BB<*D)' =4@BB?BB%()?)?D}' =.E7 BBBBBWM 0.00034 4.07407E-6 L 0.25034 4.07407E-6 *3>*B ppt_xB ppt_y=@0BBAApBB>B<*D' =4@BB?BB%()?)?D"' =.7 BBBBB'M 0 0 L 0.25 0 E*3>*B ppt_xB ppt_y=0BB<* D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(DY' =4@BB BB%()))D' =-g6B fade*<3<*(D' =1:Bhidden*o3>+B#style.visibility<*(%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(DY' =4@BB BB%()))D' =-g6B fade*<3<*1D' =1:Bhidden*o3>+B#style.visibility<*1%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =%(D' =%(DC' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*J%(D' =-6B#blinds(vertical)*<3<*JD2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*J%(D' =%(D' =%(DC' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D' =-6B#blinds(vertical)*<3<*SD2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*S%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*E%(D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*M%(D' =-6B'blinds(horizontal)*<3<*MD2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*M%(D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-6B'blinds(horizontal)*<3<*PD2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*P%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*W%(++0+80 ++0+90 ++0+:0 ++0+;0 ++0+<0 ++0+W0 ++0+X0 +N  0 ,,22F,(  ^  6u fu ^  6s s ^  6  ^  6    B ?"6@ NNN?ND"D  B ?"6@ NNN?No Uo A  <(6 ?"6@ NNN?N I ,$ 0 csComponents of internal forces resultants Swx , Swy , Swz and Mwx , Mwy , Mwz are called cross-sectional forcesXt>8 d2(CCBJBJBJBBCBBJBJBJB CCt*$v   <6 ?"6@ NNN?NM4,$ 0 FIn 3D vectors of internal forces resultants have three components each6G>0 d2FCCz  :    9,$D  02   H= jJ?"6@ NNN?N"2   N= jJ?"6@ NNN?N/: 2  B Ne jJ?"6@ NNN?N   B ?"6@ NNN?NY:Y  B ?"6@ NNN?N ~ /  B(6 Ԕ?"6@`NNN?N   ,$D 0 KSw{ Swx , Swy , Swz }>8 d2BJBBJBJBJBBCH=  B@)6 Ԕ?"6@`NNN?N  ,$D  0 YMw{ Mwx , Mwy , Mwz }>8 d2BJB@BH@BH@BH@@A>z     ,$D 0B  <D?"0@NNN?N!B  BD?"0@NNN?Np p B  BD?"0@NNN?N^c   <d6 ?"6@ NNN?Nsm  Cx&>0 d2B  <c6 ?"6@ NNN?N `  Cy&>0 d2B  <6 ?"6@ NNN?Na> Cz&>0 d2BUz %   E ,$D 0B  HciDjJ?"0@NNN?NjB  HDjJ?"0@NNN?NB  HDjJ?"0@NNN?N~pd1  B f6 Ԕ?"6@ NNN?N% ySwz@>8 d2@HA1  BL6 Ԕ?"6@ NNN?N  ySny@>8 d2@HA1  BP6 Ԕ?"6@ NNN?NYy ySwx@>8 d2@HA~z    )=) /,$D 0B ! HTDԔ?"0@NNN?N<< " B6 Ԕ?"6@ NNN?N SwL>8 d2DLEz  :  # a ,$D 02 $ H= jJ?"6@ NNN?N"2 % N= jJ?"6@ NNN?N/: 2 &B Ne jJ?"6@ NNN?N Uz    '  ,$@  0B ( HciDjJ?"0@NNN?NMB ) HDjJ?"0@NNN?NB * HDjJ?"0@NNN?Ngvxj1 + B06 Ԕ?"6@ NNN?N  yMwz@>8 d2@HA1 , B@%6 Ԕ?"6@ NNN?N` yMwx@>8 d2@HA1 - B66 Ԕ?"6@ NNN?N  yMwy@>8 d2@HAz  . },$D  0B / HTDԔ?"0@NNN?N2" 0 Bi6 Ԕ?"6@ NNN?N jMwL>8 d2DLEB 1 HTDԔ?"0@NNN?N3 2 TdD6_ж_ж?"` LCross-sectional forcesFH  0޽h ?O@  %& 33  ___PPT10 .[R+.D' = @B DB' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D ' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-g6B fade*<3<*'D)' =4@BB BB%(D' =-g6B fade*<3<*.D' =1:Bhidden*o3>+B#style.visibility<*.%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(++0+0 ++0+ 0 ++0+0 ++0+0 +XX  0 ??TTm>(  ^  6u fu ^  6u %u ^  6  ^  6    B ?"6@ NNN?Nv  B ?"6@ NNN?N    <6 ?"6@ NNN?N$ 3,$ 0 nIn 2D number of cross-sectional forces is reduced, because loading and bars axes are in the same plane (x, z):Ro>0 d2hCBCC   B ?"6@ NNN?NY:Y   B ?"6@ NNN?N     B 6 o?"6@ NNN?NG ,$D 0 Sw{ Sx , 0, Sz }>8 d2BJBBJBJBBC6   B6 o?"6@ NNN?N? ,$D  0 Mw{ 0, My , 0 }>8 d2BJBBJBBCz E    i g ,$D 0  <6 ?"6@ NNN?NE "  Cx&>0 d2BB  BD?"0@NNN?N  B  BD?"0@NNN?Nqv B  B|h{D?"0@NNN?N$ %   <6 ?"6@ NNN?N   Cy&>0 d2B  <H6 ?"6@ NNN?Nk Cz&>0 d2B"z  .  W ,$D 0 N 7 3   7 3 p N 7 3   7 3 2  HjJ?"6@`NNN?N 2  HjJ?"6@`NNN?NB  <Do?"0@NNN?N v B  <Do?"0@NNN?N  B  BDjJ?"0@NNN?N  T    C#  ~2  H jJ?"6@ NNN?N  2  H jJ?"6@ NNN?N  T    # 3z2   B o?"6@ NNN?N  2 ! B o?"6@ NNN?N  T    "C# 7  2 # H jJ?"6@ NNN?N  2 $ H jJ?"6@ NNN?N  T    %#  J 2 & B o?"6@ NNN?N  2 ' B o?"6@ NNN?N  r ( N"GB jJ?"6@ NNN?N #B )B H)\DԔ?"0@NNN?N  * BP6 ?"6@ NNN?N  GP*>0 d2F N  u   + / !r , NAG2 jJ?"6@ NNN?N  B - BDo?"0@NNN?N %  B . BDo?"0@NNN?N  B / BDo?"0@NNN?N B 0 BDo?"0@NNN?Ng n B 1 BDo?"0@NNN?N a B 2 BDo?"0@NNN?NI| P: B 3 BDo?"0@NNN?N`  B 4 BDo?"0@NNN?N+: 2 B 5 BDo?"0@NNN?N  B 6 BDo?"0@NNN?N   B 7 BDo?"0@NNN?N{   8 <6 ?"6@ NNN?N u 1  Gq*>0 d2FN Q : 9 v.1N Q 0 : : Q 0 :!N 0 sZ  ; Q AN 0 sU  < 0 sU B =B 6D?"0@NNN?N0 s4 2 > T" 1?"6@`NNN?N hU  ? <Hw6 ?"6@ NNN?N TZ  C.&>0 d2CB @ BDԔ?"0@NNN?N^0 : A <<, ?"6@ NNN?Nr  GM*>0 d2FN X d B xN Xf d C Xf d2 D T% Ԕ?"6@`NNN?N"f I 2 EB N Ԕ?"6@`NNN?NX ; 2 FB Nm Ԕ?"6@`NNN?N d G <`, ?"6@ NNN?N   GM*>0 d2Fz    H 2@ 1f ,$D 0B I BDԔ?"0@NNN?N E >N   J  B KB BDԔ?"0@NNN?N . L N!, ?"6@ NNN?N  jSx2>0 d2@H. M ND], ?"6@ NNN?N{   jSz2>0 d2@HPz  $-  N  , ,$@ 0B O BDԔ?"0@NNN?N Y  P Na, ?"6@ NNN?N $-  PMy2>0 d2@H_ Q <e, ?"6@ NNN?N 1 ,$  0 ?We will use following notations and names for these components:&@>0 d2@C{ R B|,o?"6@ NNN?N  ,$D  0 GSx=N - axial forces Sz=Q - shear force My = M - bending momentH>8 d2BJBB CJBJBB CBJBBC$/B S BDԔ?"0@NNN?N Z" ,$D 0 T Tx,_ж_ж?"` LCross-sectional forcesFH  0޽h ? !&'#$ > D E F 33[S___PPT103.[R+D' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*N%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*S%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Q%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*R%(++0+0 ++0+ 0 ++0+ 0 ++0+Q0 ++0+R0 + a  0 --44,(  ^  6u fu ^  6u %u ^  6  ^  6    B ?"6@ NNN?ND"D  B ?"6@ NNN?N  g  <lL, ?"6@ NNN?N o,$ 0 7Special cases of internal forces reductions are called:68>0 d27CC   B ?"6@ NNN?NZ=Z   B ?"6@ NNN?N^ L^ *   <, ?"6@ NNN?N E,$ 0 LTENSION  when internal forces reduce to the sum vector only, which is parallel to the bar axisr`>0 d2C&CCC CC0   <0, ?"6@ NNN?Nq` ] ,$ 0 RSHEAR  when internal forces reduce to the sum vector only, which is perpendicular to the bar axisrc>0 d2C"CCCCC:   <x7, ?"6@ NNN?N q3  ,$  0 \BENDING  when internal forces reduce to the moment vector only, which is perpendicular to the bar axisrh>0 d2 C#CCCCC  <ܸ ?"6@ NNN?N { ,$ 0 4TORSION  when internal forces reduce to the moment vector only, which is parallel to the bar axisTc>0 d2 C#CC0CJz %  :c,$D 02  H o?"6@ NNN?NB B <Do?"0@NNN?N=0B B <Do?"0@NNN?N>1B B BDo?"0@NNN?N%B  BDԔ?"0@NNN?Nnn,$@ 0B  HZDԔ?"0@NNN?N2 ,$@  0B  H+DԔ?"0@NNN?N  ,$D 0B  BDԔ?"0@NNN?N  ,$D 0Jz %  99 ,$D  02  H o?"6@ NNN?NB B <Do?"0@NNN?N=0B B <Do?"0@NNN?N>1B B BDo?"0@NNN?N%Jz %  " 9 ,$D  02  H o?"6@ NNN?NB B <Do?"0@NNN?N=0B  B <Do?"0@NNN?N>1B !B BDo?"0@NNN?N%Jz % "  9,$D 02 # H o?"6@ NNN?NB $B <Do?"0@NNN?N=0B %B <Do?"0@NNN?N>1B &B BDo?"0@NNN?N% ' <l| ?"6@ NNN?N N(>0 d2C ! ( <~ ?"6@ NNN?N } ,$ 0 CM&>0 d2B@ ) < ?"6@ NNN?N 4 ,$ 0 bMsD>0 d2BJB! * <O ?"6@ NNN?Ni(c ,$  0 CQ&>0 d2B! + <0 d2B2 , B ?"6@ NNN?N z   -  ,$@ 02 . H Ԕ?"6@ NNN?N$  @2 /  B ENGcIMQ Ԕ? J g3 `TJ g3 `TM`Tg3 `TM`T"6@ NNN?N @2 0B  B+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(De' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(De' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(+0+0+0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ ++0+(0 ++0+)0 ++0+*0 ++0++0 +`  0 e+]+::*(  =z RP  RP ,$D 0B  <D?"0@NNN?NR PT _ # S- B  <D?"0@NNN?N  <xո ?"6@ NNN?N_o En(>0 d2Cz + h  "_ ,$@ 02  NV 1?"6@ NNN?N+ B   BDo?"0@NNN?N  B   BDo?"0@NNN?NX h2   N 1?"6@`NNN?NK .    <ط ?"6@ NNN?N   EA(>0 d2C   <N ?"6@ NNN?N`? 9  bP1D>0 d2BJB.  <% ?"6@ NNN?NH7B |PnD>0 d2BJB  <X ?"6@ NNN?N  AI$>0 d2@b  BL?"6@ NNN?N]> S,$D 0z .;  5.k,$D 0B  HDo?"0@NNN?NQB  HDo?"0@NNN?N.BoB  HDo?"0@NNN?NvfB  HDo?"0@NNN?NLB  HDo?"0@NNN?NBB  HDo?"0@NNN?N;B  H Do?"0@NNN?N5z     ,$D 0N  < ?"6@ NNN?Nz  {wI}Z>0 d2DDLDr  TbGH) o?"6@ NNN?Ndz ,E   ,p ,$D 0  <dB ?"6@ NNN?N& E  j{ZI}J>0 d2@@H@r  HQ o?"6@ NNN?N,d   6( S" 8   B   0 D̙>Z,$D 0 ! 0 Dr_ ,$  0 \Neighbourhood of point A(>8 d2CB " s *D$ ,$D   06z ~ c  # p U ,$@  02 $ H= jJ?"6@ NNN?Nic 2 % N= jJ?"6@ NNN?N~  2 &B Ne jJ?"6@ NNN?N~ f2 ' 6̙JB ( 0D>],$@ 0L ) 0  ,$ 0 VThe sum of all internal forces acting on AF,>0 d2)Cc`X * 0  ,$ 0 DA  area of point A neighbourhoodd#>0 d2c`c`c + 0w,$ 0 pw6>0 d2c`B , s *D3  ,$D  0B - s *D,$D 0 . s *A ? ? L !H $D 0  / 0@ t,$ 0 rStress vector at AD>0 d2CCC 0 6o< C ,$D 0z   1 8X ,$D   0`B 2 0DjJp 3 0  An$>0 d2@z  4 e,$D 02 5 H ?#" `H  6 < ?"K ;A 2Cz 4 =  7 5L,$D  0B 8 HD̙o?"0@NNN?Nqi B 9 HD̙o?"0@NNN?N4 3B : HD̙o?"0@NNN?N* _s = H  0޽h ?/ %& 3315)5___PPT10 5.ޝP+FrD3' @= @B D3' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =%(D' =%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-s6Bwipe(down)*<3<*"D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-g6B fade*<3<*7D' =%(DK' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-s6Bwipe(down)*<3<*1D' =%(DX' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-s6Bwipe(down)*<3<*,D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =%(D<' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-6B+checkerboard(across)*<3<*-D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-o6Bdissolve*<3<*0++0+!0 ++0+)0 ++0+*0 ++0++0 ++0+/0 +=  0 eB]B\^A(    BA ? ? fQ H $D 0 B  <D?" ,$@  0]  0 ,$ 0 iStress vector is a measure of internal forces intensity and depends on the chosen point and cross sectionDj>0 d2CC3C  BA ? ?[H $D 0 z _ ^   VUz,$D 0  0= ^ Px22>0 d2@H`B  0=D_ZB   s *DF tZB   s *Di   0Țjh Px12>0 d2@H   0& =#  Px32>0 d2@Hz      ,$D 0`  0f fB  6=DjJ   0.v   Pn12>0 d2@Hz %   % ,$D  0`r  0-% `B  0DjJ5y5?  0/ Pn32>0 d2@Hz u   u ,$D  0lr  <G3-% `B  0DjJ    0(4 u  Pn22>0 d2@H6z ]B   ,$D 02  H= jJ?"6@ NNN?NH B+2  N= jJ?"6@ NNN?N] 2 B Ne jJ?"6@ NNN?N] f2  6 )   09 ,$ 0 Xp2:>0 d2DL  0H?z ,$ 0 Xp3:>0 d2DLB   0Du,$@' 0B ! s *D,$@+ 0B " s *D ,$@) 0z l  # l ,$D& 0`B $ 0DlklfB % 6Dkql `B & 0D``b`B ' 0Db`B ( 0DdiB ) 0DjJ`,$D 0B * 0D  ,$D 0B + s *D9 ,$D# 0B , s *D s ,$D! 0z C^  - C^ ,$D 0fB . 6D] ^ `B / 0D ] `B 0 0DR  R `B 1 0D CV ! `B 2 0D V B 3 0DjJ R ,$@ 0B 4 0D o ,$D 0B 5 s *Dyy ,$D 0B 6 s *D y ,$D 0z   7  ,$D 0`B 8 0D > fB 9 6D@  `B : 0DDq D `B ; 0D|Di `B < 0D| > B = 0DjJo |D ,$@ 0  > 0 C"?Mn ,$ 0 Xp1:>0 d2DL ? 0x u ,$ 0 V114>0 d2`H @ 0~& 6F ,$ 0 V124>0 d2`H A 0g2 ,$ 0 V134>0 d2`H B 0` V 6 ,$D  0 V214>0 d2`H C 0 = ,$" 0 V224>0 d2`H D 0 W ,$$ 0 V234>0 d2`H E 0p)I,$( 0 V314>0 d2`H F 0S s,$* 0 V324>0 d2`H G 0!!^A,$, 0 V334>0 d2`H H 00 d2@H@`H@ ` H@`H@H I 0W`,$% 0 *p2[21 , 22 , 23 ]>0 d2@H@`H@ ` H@`H@H J 0e`9,$- 0 *p3[31 , 32 , 33 ]>0 d2@H@`H@ ` H@`H@H K 04  ,$  0 O Point A image&>0 d2CB L s *D v ,$D 0T M 0 @ ,$. 0  11 , 12 , 13>0 d2`H@` H  @ `HT N 0L @ ,$/ 0  21 , 22 , 23>0 d2`H@` H  @ `HT O 0T8 @/ ,$0 0  31 , 32 , 33>0 d2`H@` H  @ `HT P 08 @/ ,$1 0  31 , 32 , 33>0 d2`H@` H  @ `H^z 3 2  Q {b ,$D2 0fR R 6o3 k2 fR SB 6os *  T 0  ,$3 0 TF>0 d2AiaJ U 0  {,$5 0 T(ij)r>0 d2C k c k c  V 0tb ,$ 0 QStress vectors:&>0 d2C W 0P+B ) ,$4 0 O Stress matrix&>0 d2C X 0h imP,$6 0 g i,j = 1,2,3& >0 d2 C  Y 6o P,$D7 0 Z 6Ĉo iY,$D8 0 hComponents sij of matrix T are called stresses. Stress measure is [N/m2] i.e. [Pa]S>8 d2 C7CK C ErB [ <D?"  ^ 6P S" 8   H  0޽h ?/  33x___PPT10X.ޝP+jOD||' @= @B D7|' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(Ds' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(Dc' =%(D ' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*[%(D|' =%(D$' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*K%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-o6Bdissolve*<3<*LD' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*=%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*>%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-o6Bdissolve*<3<*7D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*A%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*V%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-o6Bdissolve*<3<*-D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*C%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-o6Bdissolve*<3<*#D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*E%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*F%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*G%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*J%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*M%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*N%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*O%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*Q%(D' =-o6Bdissolve*<3<*QDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*W%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*U%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*Y%(D' =-o6Bdissolve*<3<*YDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Z%(++0+0 ++0+0 ++0+0 ++0+>0 ++0+?0 ++0+@0 ++0+A0 ++0+C0 ++0+D0 ++0+E0 ++0+F0 ++0+G0 ++0+H0 ++0+I0 ++0+J0 ++0+K0 ++0+M0 ++0+N0 ++0+O0 ++0+P0 ++0+T0 ++0+U0 ++0+V0 ++0+W0 ++0+X0 ++0+Z0 +  0 |-t-GI -(  R  6Gff> _ ,$@  0b  6%33>+ ,$@ 0  s *A ? ?JTZ H $D 0 b  633 +,$@ 0b @ 6;?  ,$@ 0b @ 6] . " ,$@ 0z M;    P> ,$D 0   0  G; g  Px22>0 d2@H   0M   Px12>0 d2@H   0%w Px32>0 d2@H`B   0=D/R ZB   s *D<{ZB  s *D{ {B  0D'  ,$@ 0B  0D ,$@ 0B @ 0D | ,$@ 0  0x  ,$ 0 Pn22>0 d2@H  0n m ,$ 0 Pn12>0 d2@H  0t)&F,$ 0 Pn32>0 d2@H  @  # 72 ,$D 0fB B 6Do  ZB B s *Dx x@ ZB B s *D    @  # J ,$D 0fB B 6Do  ZB B s *Dx x@ ZB B s *D    @  # 3 (| ,$D 0fB B 6Do  ZB B s *Dx x@ ZB  B s *D   ! 0;w /^ ,$ 0 rA1P>0 d2ckc " 0Az `a ,$ 0 rA3P>0 d2ckc # 0\G P ,$ 0 rA2P>0 d2ckcz 7  $ 7 ,$D 0`B % 0Do&7 uZB &B s *Do `B 'B 0hDo4  ( s *A ? ?R p 8 $D  0 ) s *A ? ?/m8 $D 0R * <GGGC"?y68 ,$D 0B + 0DjJ,$@ 0b , 0}E ,$ 0 (ai )>0 d2dd  @  -# ;0 ,$D  0fB .B 6Do  `B /B 0DjJx x@ `B 0B 0DjJ   1 0A ? ?C"?j ;i 8 $D  0B 2 0DԔ  ,$@ 0$ 3 0 ,$ 0 AT>0 d2gld 4 s *A ? ?v+ H $D  0  5 0N,$! 0 r SX1= 0J>0 d2 6 s *A ? ?a8 $D$ 0 7 6"`P,$@# 0 8 s *A ? ?VA8 $D" 0 9 s *A ? ?M 78 $D* 0 : s *A ? ?74'8 $D( 0 ; s *A ? ?( 0 8 $D' 0 < 05,$) 0 N & , & &>0 d2C = 0"`>J,$@+ 0 > s *A ? ?(8 $D, 0 ? s *A ? ?N x8 $D- 0 @ s *A ? ? #; 8 $D/ 0 A s *A ? ?]  8 $D1 0 B s *A ? ? 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The coordinate system defined by the directions of principal stresses is called system of principal axis.&/>0 d2/Cz  u  A 1 F ,$D 0f B 6f  fB C 6DjJ  fr D 6 fG fB E 6^lDjJL<L lr F <^lG3e" fB G 6DjJ H BA ? ?u6  I BA ? ?J  J BA ? ? 7  z 0 K xw_,$D   0 L BA ? ?0.  [`B M 0DԔ#Cxz R  N i ,$D   0 O BA ? ?TRg  `B P 0DԔ  xz  o Q ; K,$D  0 R BA ? ?  `B SB 0DԔ o T 0z G H ,$D  0 z  7 U  7,$D 0 V 6Uf[f   O22>0 d2@H`B W 0/o Dl `B X 0D `B Y 0UD$8 W8  Z 6U@l[7 O12>0 d2@H [ 6Uhq[  O32>0 d2@H \ BA ? ?"`Y 8 $D 0 ] BA ? ?"`d7 8 $D  0 ` 6u[ S" 8  [ H  0޽h ? 33++___PPT10t+.Q >-+-DH+' @= @B D+' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@/%(D'  =-o6Bwipe(up)*<3<*@/D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =-s6Bwipe(left)*<3<*TD4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*U%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*A%(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Q%(D' =-o6Bwipe(up)*<3<*QD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*N%(D' =-s6Bwipe(left)*<3<*ND' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*K%(D' =-s6Bwipe(down)*<3<*KD' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*]%(DP ' =%(D ' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*AD' =1:Bhidden*o3>+B#style.visibility<*A%(D)' =4@BB BB%(D' =-g6B fade*<3<*KD' =1:Bhidden*o3>+B#style.visibility<*K%(D)' =4@BB BB%(D' =-g6B fade*<3<*ND' =1:Bhidden*o3>+B#style.visibility<*N%(D)' =4@BB BB%(D' =-g6B fade*<3<*QD' =1:Bhidden*o3>+B#style.visibility<*Q%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*T%(+' 0 P (  Zz F;  F;,$D  0Z  0t=[J; It can be proved that are extreme values of normal stresses (stresses on a main diagonal of stress matrix) . Customary, these values are ordered as follows&>0 d2Cn  s *A! ? ?F| <   0A" ? ?C"?/ 8 $D 02  6o  ,$D   02  6o ,$D   02  6o X ,$D  0z M ( C     M 3 ,$D 0fB   6DoM < <    0A# ? ?S"`? ( C  z F ? <     F ? , ,$D 0fB   6DoM V <   0A$ ? ?S"`?F ?  z  < p  ,  ` ,$D 0fB B 6Do < R d  0A% ? ?S"`? E p   0H[z@ ,$  0 Surface of an ellipsoid with semi-axis (equatorial radii) equal to the values of principal stresses represents all possible stress vectors in the chosen point and under given loading. &>0 d2CrB  <D?"8 P P 8   6pO[ S" 8 DInternal forces  state of stress ,#d!Fx  B ?"FH  0޽h ? 33___PPT10j.Q >-+_D' @= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(Dc' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-o6Bwipe(up)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D ' =%(D' =%(Dp' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-o6Bwipe(up)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*+8+0+0 +  0 ""p55k"(   5 <?"&,$D 0  s *S" ? ,$D  0  s *S" ? ,$D!  0  s *S"`?2,$D"  02  0f"` ,$D(  0  s *A& ? ?"`  8 $D  0  BA' ? ?"` 3 8 $D  0  s *A( ? ?"` 8 $D 0   BA) ? ?"` J 8 $D 0   s *A* ? ?"` r 8 $D 0   BA+ ? ?"` R 8 $D 0   s *A, ? ?"`  8 $D 0   s *A- ? ?"` 8 $D 02  6Ԕ +,$D 0  0h[ 9,$@ 0 z max t@>0 d2G 2  0! oY ,$@ 02  0& ! ^ ,$@ 02  0# [ ,$@  0RB  s *D _ _ 2  0a,$D 02  01 i ,$D 02  0 2 ,$D 0B  0D ,$D 0B  0D : ,$D  0  s *A. ? ?"`2 7 8 $D 0  s *A/ ? ?"` 8 $D 02  s *"` ,$D&  02  s *S"`?w  ,$D$  0Zz     ,$D 0`B  0DjJ% % |  s *A0 ? ?"`   |   s *A1 ? ?"`-  `B ! 0DjJB YB B " 0D{ {,$D 0 # 0[:,$1 0 "Mohr circles  represent 3D state of stress in a given point  on the plane of normal and shear stresses4j>8 d2 A]C2 $ 0S"`?w  ,$D#  02 % 0"` ,$D%  02 & 0"` ,$D'  0 ' s *S" ?B *,$D)  0 ( 0A2 ? ?"`3 } 8 8 $D 0B ) 0DT T B ,$D 0B * 0D~: ,$D 0 + s *A3 ? ?"`@8 $D, 0 , s *A4 ? ?"` S \ 8 $D0 0 - s *A5 ? ?"`  8 $D. 0 . s *A6 ? ?"`8 $D- 0 / s *A7 ? ?"`i8 $D+ 0B 0 0D  : ,$D 0 1 s *A8 ? ?"` ) 2 r 8 $D/ 0 2 s *A9 ? ?"`> S 8 $D  0 3 0h[ ,$D 0 W0&>0 d2C  4 S 0A: ZWM-LOGO"` 4  ,$D2 0H  0޽h ? 33tt___PPT10_t.Q >-+tDs' $[= @B DFs' = @BA?%,( < +O%,( < +D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-g6B fade*<3<*5D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(DY' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bwipe(right)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-s6Bwipe(down)*<3<*$D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-s6Bwipe(down)*<3<*%D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-s6Bwipe(down)*<3<*&D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-s6Bwipe(down)*<3<*'D' =%(D' =%(D3' =4@BB*BB%()?)?D' =.O7 BBBBBaM -5.55556E-7 2.96296E-6 L -5.55556E-7 0.24722 *3>*B ppt_xB ppt_y=@0BBAApBBB'=<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(++0+0 ++0+#0 ++0+30 +O 0 !!(  2  Nvyd @"  ,$D  02  Hvyd @"  ,$D  0  HA; ?Ԕ ?"` $8 $D 0B  6DԔKK,$D  0z f _   #,$D 0   NBCEDEFBZf((o&2;NH Fn,Y2U|jFJv0YEt@c1ap[o @             `"<   \ O    HBCEDEFB((o&2;NH Fn,Y2U|jFJv0YEt@c1ap[o @             `"T  ?f `B   0D!`B   0D. . `B   0D  _ `B   0D   `B   0D B `B  0DR R eLz h    !h,$D 0&T [e  # ~h  TB  c $D[ fB  6D& e   BA< ? ?"`  u !  0 K> ,$  0 KOn the body surface stress vector has to be balanced by the traction vector&L>0 d2LC  BA= ? ?g,*H $D  0 >B  6DԔ<<,$D   0  BA> ? ?rq8 $D  0  HA? ?Ԕ ?S"`?pv8 $@   0  0Q>w ^,$ 0 \Stress on the body surface&>0 d2C  0W>  ,$ 0 m+Coordinates of vector normal to the surface&,>0 d2,C  0C"?,$D  0B @ s *DE+v,$D 0B  s *D#*,$D 0  HA@ ?Ԕ ?S"`?v @ 8 $D  0S  0\>  ,$  0 ]This equation states statics boundary conditions to comply with the solution of the equation:F^>0 d2CG.C   0c> ?,$  0 {This equation (Navier equation) reflects internal equilibrium and has to be fulfilled in any point of the body (structure).f|>0 d2CG CG?C ! 6l> S"   TStress distribution $dBH  0޽h ? 3366___PPT10j6.Q >-+=u D&5' >= @B D4' = @BA?%,( < +O%,( < +D ' =%(D' =%(Dp' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D+' =4@BB?BB%()?)?D' =.G7 BBBBBYM 2.77778E-7 -3.7037E-6 L 0.08993 -0.01921 *3>*B ppt_xB ppt_y=@0BBAApBB-8=BI<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-o6Bwipe(up)*<3<*D' =%( D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-o6Bwipe(up)*<3<*D' =%( D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(Ds' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-o6Bwipe(up)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D'  =-o6Bwipe(up)*<3<* ++0+0 ++0+0 ++0+0 ++0+0 ++0+ 0 +  0    0 (    HA; ?Ԕ ?"` 8 $D 0  0̏>p ,$  0 HWe have to deal with the set of 3 linear partial differential equations.&I>0 d2IC  0>{h,$ 0 QNavier equation&>0 d2C   0P>@h',$ 0 xin coordintes reads:&>0 d2C   HA ?Ԕ ?"` 8 $D 0*  0>a  ,$  0 TThere are 6 unknown functions which have to fulfil static boundary conditions (SBC):&U>0 d2UC  HAB ?o ?% i  H $D 0 >[   0t>Q ,$   0 We need more equations to determine all 6 functions of stress distribution. To attain it we have to consider deformation of the body.&>0 d2C   6> S"   \Stress distribution ,dFH  0޽h ? 33___PPT10.Q >-+E\dDd' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D ' =%(D' =%(Dp' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* ++0+0 ++0+0 ++0+0 ++0+0 ++0+ 0 +  0 *)EE{)(  z ZC  + ,$D 02  0S" ?b C `B  0D3Ԕi Zj X `B  0D3Ԕ 2  0"`&  V j ,$D* 0  0>  % ,$ 0 AA$>0 d2@z  m    > ,$@ 0`B   0D : ZB   s *D9 9 ZB   s *D" m B   s *D?  4,$D  0   0<> .  ,$% 0 DA $>0 d2@2  0"`5 d y ,$D# 0B  s *D ] J,$@$  0  BAC ? ?"`U [8 $D 0  BAD ? ?"`D G3 8 $@/ 0  6>3Ԕ# ,$D  0 UDisplacement vector&>0 d2C2  s *3S" ?O) ,$D 02  0S" ?;Ar,$@  0B  0D3Ԕ,$@  0B  0D3Ԕ  ,$D   0B  0D3Ԕ;9,$D 0B  0D3Ԕca ,$@ 02  0"`uW,$@ 0  0>4L:T,$ 0 AA$>0 d2@z  m   pB ,$D 0`B  0D : ZB  s *D9 9 ZB  s *D" m B  s *DU0 ,$@  0   0>, L,$ 0 DA $>0 d2@ ! BAE ? ?)/8 $D 02 " 0"`uT,$@ 0B # s *D"Yd ,$D  0B $ 0D3Ԕ,d*,$D  0 % BAF ? ?Q8 $D 0 & BAG ? ?"`s8 $D 0 ' 0> ~ l ,$( 0 AB$>0 d2@2 ( 0"`*  X n ,$@' 0 ) 0`>N / n ,$+ 0 DB $>0 d2@ -C *# EYs c i ( ,$D&  0ZB + s *D>wCwTB , c $D-TB - c $D;-;z  $ G  .  J  ,$D,  0`B / 0D>  ZB 0 s *D  G ZB 1 s *D$  2 2 0"`7 g { ,$@  0B 3@ s *DL Q ,$D2 0B 4@ s *D E K,$D3 0 5 BAH ? ?"`D y8 $D 02 6 $B[hENGMIQ [hf[h`T[hf[h`T`Tf[h`T`T X ,$@4 0 7 BAI ? ?"`  8 $@6  0 8 BAJ ? ?"` 8 $D 0 9 0 > v ,$5 0 Da&>0 d2z y f    :  * ,$D-  0`B ; 0Doy 8   < BAK ? ?"` f   z F RS  = / 5 A ,$D.  0`B > 0DoF S  ? BAL ? ?"`" R   @ BAM ? ?"`#  8 $D 0B A s *Da a ,$@0 0B B s *D  ,$@7 0B C s *D ZG ,$D8 0B D s *D^ ^ ,$D1 0 E 6h> S"  gState of deformation - strains , dFH  0޽h ?6 33;3___PPT10.'s+=RD' >= @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(Ds' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*DL' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(Dg' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D+' =4@BB?BB%()?)?D' =.G7 BBBBBYM 3.61111E-6 -4.81481E-6 L 0.05434 0.03426 *3>*B ppt_xB ppt_y=@0BBAApBB<BpS<<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-s6Bwipe(down)*<3<*#D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bwipe(up)*<3<*$D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(Dg' =%(D' =%(D+' =4@BB?BB%()?)?D' =.G7 BBBBBYM -3.61111E-6 -3.7037E-7 L 0.03507 0.07801 *3>*B ppt_xB ppt_y=@0BBAApBB<BI=<*2D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D(' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-s6Bwipe(down)*<3<**D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(Dc' =%(D ' =%(D'' =4@BB?BB%()?)?D{' =.C7 BBBBBUM -8.33333E-7 3.33333E-6 L 0.0809 0.0956 *3>*B ppt_xB ppt_y=@0BBAApBB[%=BQC=<*(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-s6Bwipe(down)*<3<*.D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =-o6Bwipe(up)*<3<*:D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*=%(D' =-o6Bwipe(up)*<3<*=DL' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*A%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-u6Bwipe(right)*<3<*3Df' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-u6Bwipe(right)*<3<*4Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9%(D/' =%(D' =%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-s6Bwipe(down)*<3<*7D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*C%(++0+0 ++0+0 ++0+ 0 ++0+0 ++0+0 ++0+ 0 ++0+'0 ++0+)0 ++0+90 +1 0   G (    HAN ?o ?m^8 $D 0  0 : ,$  0 O Small strains&>0 d2C>  0:u UH ,$D  0 ,Shear strains when ij@>0 d2C@  0:m,$D 0 sNormal strains when i=j@>0 d2C@  HAO ?o ?C"? ! { 8 $D  0  HAP ?o ?C"?`]8 $D 0  HAQ ?o ?C"?{8 $D 0   HAR ?o ?C"?}/ 8 $D  0   HAS ?o ?C"? a 8 $D   0   HAT ?o ?C"?F 1 8 $D  0   HAU ?Ԕ ?C"?, 8 $D   0R   0:  ,$D  0 zStrain matrix  symmetrical by definition of angular strains(>>8 d2>C  NA ? ?#" `_H $D 0 :  6%: S"  gState of deformation - strains , dFH  0޽h ? 33""___PPT10".'s+vrD!' = @B D|!' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D'  =-o6Bwipe(up)*<3<* D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* ++0+0 ++0+0 ++0+0 ++0+ 0 +r  0  !"(  z $}   $} ,$D 0ZB  s *D  t ZB  s *D A | ZB B s *DP P   0C:$0 ~  bx1D>0 d2CKC  0I:# }  bx2D>0 d2CKC  0N: K  bx3D>0 d2CKCB   s *Doq W ,$D  0B   s *DoA X ,$D  0B   s *Do~n q ,$@  0B   s *Dol _ ,$D  0B   s *DoS F ,$D  0B  s *Dov i,$@  0B  s *DoZ g,$@  0B  s *Doolk ,$D  0B  s *Dotf Z,$@  0B  s *DR i ,$D  0B  s *DZ @ ,$D  0B  s *D`b S ,$@  0B @ s *Do [ R ,$D  0B @ s *DoP  P ,$D 0B  s *Dod P ,$D  0B @ s *D Y U ,$D  0B  s *DoW W ,$D   0B  s *Do`L ,$@   0B @ s *D b O ,$D  0B @ s *Do Y d,$D   0B @ s *Dob e,$@   0B @ s *Do] Y ,$D  0B @ s *Dod d,$D  0B   s *D` ` ,$@   0 " 6p`: S"  gState of deformation - strains , dFH  0޽h ? 33|_t____PPT10T_.P%K+}?D(_' = @B D^' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bwipe(right)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*De' =%(D ' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D&' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*De' =%(D ' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D ' =%(DL ' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(+#1 0    (    HAU ?Ԕ ?C"?EE8 $D 0  0 :C"?/t,$  0 Eigenvalues of strain matrix are normal strains on the planes where there are no shear strains. Principal strains can be found by solving the secular equation:&>0 d2C  HAV ?Ԕ ?C"?_ 8 $D  0  HAW ?Ԕ ?C"? /8 $D  0  08:Ee ,$  0 Ffwhere I1e, I2e, I3e are invariants of strain matrix4>0 d2C@H@@H@@H@ CQ  0 :  ,$  0 kWhen transfer of strain matrix is made to the new co-ordinate system then matrix transformation rule holds:6l>0 d2kC  BA ? ? x b H $D  0 :6   0:~ Cl,$   0 `In the co-ordinates defined by principal directions of strain matrix it takes the diagonal form.&a>0 d2aCB   0O  ,$D   0   BA ? ?~ 8 $D 0   <: ?" ,$  0 K- principal strains 2CR   B?"  z,$D  0  6̪: S"  gState of deformation - strains , dFH  0޽h ? 33""___PPT10".'s+_hDh!' = @B D#!' = @BA?%,( < +O%,( < +Dt' =%(D' =%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-o6Bwipe(up)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-s6Bwipe(left)*<3<*D$' =%(D' =%(Dt' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-s6Bwipe(left)*<3<*D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D ' =%(D' =%(Dp' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D ' =%(D' =%(Dp' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(++0+0 ++0+0 ++0+0 ++0+ 0 ++0+ 0 +R  0 W(     <: ?" Z,$ 0 [Strain versus stress ( 2 C  < ?"(  ,$D  0  < ?" ,$D 0z t6   T>,$D  02  Bo?#" `6V   <: ?"t  [!Material deformability properties"( 2"C7   <x: ?"ch ,$  0 QGeneral property of majority of solids is elasticity (instantaneous shape memory)R 2RCB   s *DԔ$ ,$D   0z '     ,$D  0ZB   s *D&D& ZB   s *D    0|:'7 AP$>0 d2@  0:f   Au$>0 d2@LB  c $D hh B  s *DԔ Y,$D   0  0T:  ,$  0 SLinear elasticity&>0 d2C   <: ?" ,$  0 ^"Deformation versus internal forces #( 2#C  0:Z r.A ,$  0 PCEIIINOSSSTTUV&>0 d2C  0<:` x4G ,$  0 SUT TENSIO SIC VIS&>0 d2C  s *A ? ?"` T 8 $D 0R  <: ?" x  ,$  0 jwhich reads:  as much the extension as the force is 6( 26CP  0h=  ,$  0 <where k is a constant dependent on a material and body shapeR=>0 d2C@'CG @   <d ?" e ,$  0 r:This observation was made already in 1676 by Robert Hooke:; 2;C  6 S" ( j"Constitutive equation - Hooke law ,#d!FH  0޽h ? 33>>___PPT10>.ޝP+[DP<' = @B D <' = @BA?%,( < +O%,( < +D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-s6Bwipe(left)*<3<*#D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bwipe(right)*<3<*D' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*#D' =1:Bhidden*o3>+B#style.visibility<*#%(D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D'  =-o6Bwipe(up)*<3<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D' =%(D' =%(D1' =4@BBBB%(D' =-o6Bwipe(up)*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(D5' =4@BBBB%(D' =-s6Bwipe(down)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-o6Bwipe(up)*<3<*D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D3' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-s6Bwipe(left)*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-o6Bwipe(up)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-o6Bwipe(up)*<3<*+0+0+0 ++0+ 0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 +K  0 ;3(  B  c $DZ Z,$@  0B  c $D' ',$@  0B  c $D   ,$D  0  0TƬ`X,$   0 d Physical quantities (measurable)(!>8 d2!C  0 ʬXq ,$   0 l(Mathematical quantities (non-measurable)()>8 d2)C  0̬2t ,$   0 JDynamics& >0 d2 C  0Ϭ"#  ,$   0 L Kinematics& >0 d2 CB   c $DD ,$@  0B   c $DDVV ,$@  0B   c $DD ,$D  0   s *AA ? ?r 8 $D  0   s *A ? ? 8 $D 0  s *A ? ? > 8 $D 0  s *A ? ? , 8 $D 0  0) ,$D  0  0>^,$ 0 M Hooke, 1678& >0 d2 C  0֬^ ,$ 0 N Navier, 1822& >0 d2 C  0$\,$D  0  s *A ? ? 8 $D 0P  0\, X ,$  0 Zf - linear function of all strain matrix components defining all stress component matrix F[>8 d2BCYC:  <̵ ?"d,$D  0 TTo make Hooke s law independent of a body shape one has to use state variables characterizing internal forces and deformations in a material point i.e. stress and strain. 2C  6 S" ( j"Constitutive equation - Hooke law ,#d!FH  0޽h ? 3399___PPT108.ޝP+dD;7' = @B D6' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-o6Bwipe(up)*<3<*De' =%(D ' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*DY' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-o6Bwipe(up)*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-o6Bwipe(up)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-o6Bwipe(up)*<3<*++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 +T  0 f(  ]  0xL2[,$  0 gThe coefficients of this equation do depend only on the material considered, but not on the body shape.Fh>0 d2MCGC  00 d2CKC  BA[ ? ?;+ 8 $D 0y  0t[@ 2 ,$  0 SSummation over kl indices reflects linear character of this constitutive equation. TT>0 d2 GCBCCC  6`c S" ( j"Constitutive equation - Hooke law ,#d!FH  0޽h ? 33&  ___PPT10 .O=+:D* ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*++0+0 ++0+0 ++0+0 +\; 0 " (    HA\ ?o ? uH $D 0   HA] ?o ?<^8 $D 0  BGyeHJ NpeK,$D 0 WSummation obeys !*> dG  BGH N C,$D 0 kKronecker delta&> dC 7  00Y+,$ 0 Lam constants [Pa]Z>0 d2Cc C c   HA^ ?o ?S"`?t 8 $D  0  HA_ ?o ?S"`?  8 $D   0   HA` ?o ?S"`?  8 $D  0   HAa ?o ?S"`?R $S 8 $D   0   HAb ?o ?S"`? $ 8 $D 0   HAc ?o ?S"`? 8 $D 0  @ 0f ,$D   0 @ 0 $ ,$D  0  0@vh ,$  0 m+Normal stress and normal strain dependences&,>0 d2,C  0ڬ > ,$ 0 k)Shear stress and shear strain dependences&*>0 d2*C   <߬ ?"p ,$  0 ]%This equation consists of two groups:& 2&C  6 S" ( j"Constitutive equation - Hooke law ,#d!FH  0޽h ?/  33**___PPT10*.O=+D>)' = @B D(' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+P+0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 +:M 0 ph`(    HAd  ?o ? w8  $@ 0  BG |H 5,$D 0 WSummation obeys !*> dG$  BPG3HH X ,$D 0 ~"Kronecker s delta&> dC   0H S,$ 0 Poisson modulus [0]8>0 d2Cc  HAf ?o ?S"`?^ _ 8 $D  0   HAh ?o ?  c8 $D 0   0S,$ 0 Young modulus [Pa]8>0 d2Cc    HAi ?o ?S"`? l 8 $D  0   HAj ?o ?S"`? \8 $D  0   HAk ?o ?S"`?= ]L 8 $D  0  HAl ?o ?S"`? 6 8 $D 0  HAm ?o ?S"`? 68 $D 0  HAg ?o ?S 8 $D 0  HAn ?o ?S"`?  8 $D 0  HAo ?o ?S"`?"8 $D  0  0FY,$D  0  HAp ?o ?S"`? 5 8 $D 0  05_ = ,$  0 m+Normal stress and normal strain dependences&,>0 d2,C  0: H ,$ 0 k)Shear stress and shear strain dependences&*>0 d2*C @ 6jJ7 ,$D   0 @ 6jJ^ Q |,$D  0T  6? S" ( "Constitutive equation - Hooke law ,#d!FV H  0޽h ?/  33r8j8___PPT10J8.O=+ D6' = @B D6' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D%' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-s6Bwipe(left)*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-s6Bwipe(left)*<3<*+P+0+0 ++0+0 ++0+0 ++0+ 0 ++0+0 ++0+0 +63 0 g _ @ (    HAq ?o ?S"`?8 $D 0  HAr ?o ?S"`?8 $D 0  HAs ?Ԕ ?S"`?6 F8 $D 0  0diR 9,$  0 SVolume change law&>0 d2C  0n h,$  0 PDistortion law&>0 d2C  HAu ?o ?S"`? n 8 $D  0B  s *Do   ,$@   0B   s *Do T < ,$D   0   HAt ?o ?S"`?Gn W 8 $D 0   HAx ?o ?S"`? * 8 $D  0   HAy ?o ?S"`? o* 8 $D 0   HAz ?Ԕ ?S"`?W ( h8 $D 0x  B?"x  B?"  NA ? ?#" ` HuU H $D  0   6v S" ( j"Constitutive equation - Hooke law ,#d!FH  0޽h ? 33w%o%___PPT10O%.O=+ODtD$' = @B Dn$' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(Dc' =%(D ' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D&' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*+p+0+0 ++0+0 +9  0 yp(    BA{ ? ?39 ?H $D 0   BA| ? ? '8 $D 0  BA} ? ?8 $D  0  HA; ?Ԕ ?S"`?1=8 $D 0  HAB ?o ?C"?` "z8 $D 0  BA~  ? ?G3 H  $D  0   0g`N,$ 0 DNE&>0 d2C   0$`,$ 0 FCE(>8 d2C   0!H`/ ,$  0 FHE(>8 d2C   0 %G.,$ 0 GSBC(>8 d2C   0蘋,$  0 GKBC(>8 d2CB   s *DԔ h ,$D  0,  0| `I ,$  0 VThe set of NE+CE+ HE equations consists of 15 linear differential-algebraic equations  and is always the same for any static problem (except of material constants in HE).&>0 d2C  0|I hp,$  0 ^Individual problems are different only due to different boundary conditions, which define body shape ani , loading qni and displacements ui on the body surface (supports). Here is where name Boundary Value Problem of Elasticity comes from.0>0 d28CGC G GGGC$G C,se  0Dn R,$  0 E,(>0 d2C4  6 S"  `BVP Boundary Value Problem of Linear Elasticity ,1d/FH  0޽h ? 33''___PPT10&.s+iD%' = @B D$' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*++0+0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+0 ++0+0 ++0+0 +G>  0 --362-(  ^  6u fu ^  6u %u ^  6  ^  6    B ?"6@ NNN?ND"D  B ?"6@ NNN?N     B ?"6@ NNN?NZ=Z   B ?"6@ NNN?N^ L^    <Ӌ ?"6@ NNN?N E LTENSION  when internal forces reduce to the sum vector only, which is parallel to the bar axisr`>0 d2C&CCC CC   <`ۋ ?"6@ NNN?Nq` ]  RSHEAR  when internal forces reduce to the sum vector only, which is perpendicular to the bar axisrc>0 d2C"CCCCC   <H ?"6@ NNN?N q3   \BENDING  when internal forces reduce to the moment vector only, which is perpendicular to the bar axisrh>0 d2 C#CCCCC  <h ?"6@ NNN?N {  4TORSION  when internal forces reduce to the moment vector only, which is parallel to the bar axisTc>0 d2 C#CC0CF %  :c2  H o?"6@ NNN?NB B <Do?"0@NNN?N=0B B <Do?"0@NNN?N>1B B BDo?"0@NNN?N%B  BDԔ?"0@NNN?NnnB  HZDԔ?"0@NNN?N2 F %  99 2  H o?"6@ NNN?NB B <Do?"0@NNN?N=0B B <Do?"0@NNN?N>1B B BDo?"0@NNN?N%F %  " 9 2  H o?"6@ NNN?NB B <Do?"0@NNN?N=0B  B <Do?"0@NNN?N>1B !B BDo?"0@NNN?N%F % "  92 # H o?"6@ NNN?NB $B <Do?"0@NNN?N=0B %B <Do?"0@NNN?N>1B &B BDo?"0@NNN?N% ' <0— ?"6@ NNN?N N(>0 d2C  ( <ė ?"6@ NNN?N }  CM&>0 d2B  ) <Ɨ ?"6@ NNN?N 4  bMsD>0 d2BJB * <4˗ ?"6@ NNN?Ni(c  CQ&>0 d2B + <ϗ ?"6@ NNN?N*h$ CN&>0 d2B2 , B ?"6@ NNN?N F   -  2 . H Ԕ?"6@ NNN?N$  @2 /  B ENGcIMQ Ԕ? J g3 `TJ g3 `TM`Tg3 `TM`T"6@ NNN?N @2 0B  B+B#style.visibility<*5%(D'  =-o6Bwipe(up)*<3<*5D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-s6Bwipe(left)*<3<*6D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-o6Bwipe(up)*<3<*6D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-o6Bwipe(up)*<3<*6D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-o6Bwipe(up)*<3<*6+@  0 j-b-26$,(  $^ $ 6u fu ^ $ 6u %u ^ $ 6  ^ $ 6   $ B ?"6@ NNN?ND"D $ B ?"6@ NNN?N    $ B ?"6@ NNN?NZ=Z  $ B ?"6@ NNN?N^ L^   $ <H ?"6@ NNN?N E LTENSION  when internal forces reduce to the sum vector only, which is parallel to the bar axisr`>0 d2C&CCC CC  $ <䰗 ?"6@ NNN?Nq` ]  RSHEAR  when internal forces reduce to the sum vector only, which is perpendicular to the bar axisrc>0 d2C"CCCCC  $ <, ?"6@ NNN?N q3   \BENDING  when internal forces reduce to the moment vector only, which is perpendicular to the bar axisrh>0 d2 C#CCCCC $ <\; ?"6@ NNN?N {  4TORSION  when internal forces reduce to the moment vector only, which is parallel to the bar axisTc>0 d2 C#CC0CF % $ :c2 $ H o?"6@ NNN?NB $B <Do?"0@NNN?N=0B $B <Do?"0@NNN?N>1B $B BDo?"0@NNN?N%B $ BDԔ?"0@NNN?NnnB $ HZDԔ?"0@NNN?N2 F % $ 99 2 $ H o?"6@ NNN?NB $B <Do?"0@NNN?N=0B $B <Do?"0@NNN?N>1B $B BDo?"0@NNN?N%F % $ " 9 2 $ H o?"6@ NNN?NB $B <Do?"0@NNN?N=0B  $B <Do?"0@NNN?N>1B !$B BDo?"0@NNN?N%F % "$  92 #$ H o?"6@ NNN?NB $$B <Do?"0@NNN?N=0B %$B <Do?"0@NNN?N>1B &$B BDo?"0@NNN?N% '$ < ?"6@ NNN?N N(>0 d2C  ($ <x ?"6@ NNN?N }  CM&>0 d2B  )$ <TQ ?"6@ NNN?N 4  bMsD>0 d2BJB *$ <  ?"6@ NNN?Ni(c  CQ&>0 d2B +$ <8  ?"6@ NNN?N*h$ CN&>0 d2BF   -$  2 .$ H Ԕ?"6@ NNN?N$  @2 /$  B ENGcIMQ Ԕ? J g3 `TJ g3 `TM`Tg3 `TM`T"6@ NNN?N @2 0$B  B+B#style.visibility<*5$%(D'  =-o6Bwipe(up)*<3<*5$D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6$%(D' =-s6Bwipe(left)*<3<*6$D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6$%(D' =-o6Bwipe(up)*<3<*6$D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6$G%(D' =-o6Bwipe(up)*<3<*6$GD' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6$G%(D' =-o6Bwipe(up)*<3<*6$GD' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6$S%(D' =-o6Bwipe(up)*<3<*6$S+)  0 @8P(   c $+ C"?y  H  0޽h ? ̙33___PPT10i.h`P++D=' = @B +  0 ,(  ,r ,S /\  H , 0޽h ? ̙3380___PPT10.5  0 0&(  x c $$֗ =   n  Tח_ж_ж?"` (De Saint-Venant and Bernoulli hypotheses))Ft   H  0޽h ? 33___PPT10i.cp+D=' = @B +   0   @\ (    0 pB R2. Hipoteza B: Przekroje poprzeczne prta pBaskie przed przyBo|eniem obci|enia pozostaj pBaskie tak|e po jego przyBo|eniu Pozwala ona (szczeglnych przypadkw redukcji siB przekrojowych) na zaBo|enie - dla poszczeglnych przypadkw wytrzymaBo[ciowych - przemieszczeD przekroju poprzecznego prta i na tej podstawie wyznaczenie odksztaBceD a nastpnie napr|eD. Wyznaczenie przemieszczeD wymaga uwzgldnienia kinematycznych warunkw brzegowych. Hipoteza B pozwala na okre[lenie napr|eD i odksztaBceD w dowolnym przekroju prta bez potrzeby uciekania si ka|dorazowo do rozwizywania skomplikowanego zadania brzegowego TS. Hipotez B stosuje si do podstawowych przypadkw wytrzymaBo[ciowych: rozciganie skrcanie a tak|e [cinanie  przy odpowiednich ograniczeniach co do geometrii prta (prt pryzmatyczny, o dBugo[ci znacznie wikszej ni| wymiary poprzeczne)fPP2CG1CGC _j  TT_ж_ж?"` (De Saint-Venant and Bernoulli hypotheses))Fp   H  0޽h ? 33___PPT10i.cp+D=' = @B +; 0 `((  ^  S A )     c $_ .    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